Abstract
In this study, first we investigate the Fibonacci vectors, Lucas vectors and their vector products considering two Fibonacci vectors, two Lucas vectors and one of each vector. We give some theorems for the mentioned vector products and then we give the conditions for such vectors to be perpendicular or parallel. We also introduce the area formulas for the parallelograms constructed by Fibonacci and Lucas vectors with respect to Fibonacci and Lucas numbers. Moreover, we determine some formulas for the cosine and sine functions of the angles between two Fibonacci vectors, two Lucas vectors and lastly a Fibonacci vector and a Lucas vector. Finally, we investigate the Fibonacci quaternions and Lucas quaternions. We give some corollaries regarding the quaternion products of two Fibonacci quaternions, two Lucas quaternions and one of each quaternion. We conclude with the result that the quaternion product of such quaternions is neither a Fibonacci quaternion nor a Lucas quaternion.
Highlights
Fibonacci numbers are one of the most fascinating subjects of mathematics since they involve many secrets some discovered, some yet to be discovered for over 700 years since they were first introduced
Similar to Fibonacci numbers, another sequence called Lucas numbers were introduced by Edouard Lucas, a French mathematician, in 1878 and they became a widely studied subject of mathematics [8]
Since the area of a parallelogram constructed with the vectors u, v is calculated by u × v, Theorems 3.1-3.3 give us the following corollary: Corollary 3.5. (i) The area of a Fibonacci parallelogram constructed by fn1 and fn2 is Afn1 fn2 = 3Fn2−n1
Summary
Fibonacci numbers are one of the most fascinating subjects of mathematics since they involve many secrets some discovered, some yet to be discovered for over 700 years since they were first introduced. Similar to Fibonacci numbers, another sequence called Lucas numbers were introduced by Edouard Lucas, a French mathematician, in 1878 and they became a widely studied subject of mathematics [8]. The quaternions were defined by Irish mathematician Sir William Rowan Hamilton [6]. They can be considered as an extension to the complex numbers. Flaut and Savin studied Fibonacci and Lucas quaternions [3]. In the case of vectors, we consider the products of Fibonacci and Lucas vectors. We consider the Fibonacci and Lucas quaternions and their quaternion products and we give some important results
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